Non-Stochastic Processes

For example, consider the ODE:

Let $x(t)\in \mathbb{R}$ for $t\in [0,\infty )$, and

$$\frac{dx}{dt}=x,x(0)=1$$

Then the solution is to consider $x(t)=A{e}^t$. Then via $x(0)=1$, we see $x(t)=e^t$.

Discretization

We can simulate/solve the ODE by discretizing it. Let $\delta t>0$ be small and let $t_n=n\delta t$ for $n=0,1,2,\dots$. How are $x(t_n),x(t_{n+1})$ related? We have

$$x(t_n)=\frac{dx}{dt}{|}_{t=t_n}\approx \frac{x(t_{n+1})-x(t_n)}{\delta t}$$

which implies

$$x(t_{n+1})\approx x(t_n)+x(t_n)\delta t$$

Consider the difference equation:

$$x_{n+1}=x_n+x_n\delta t,x_0=1,n=0,1,2,\dots$$

For example,

$$x_1=1+\delta t,x_2=(1+\delta t{)}^2$$

such that

$$x_n=(1+\delta t{)}^n$$

showing how we can approximate the discrete solution $x_n$ to the continuous solution $x(t)$. But does it approach it?

Let $t^{\prime }=t_n=n\delta t$ and consider the limit $n\to \infty ,\delta t\to 0$ with $t^{\prime }$ fixed

Then

$$x_n={\left(1+\frac{t^{\prime }}{n}\right)}^n\to e^{t^{\prime }}=x(t^{\prime })$$

as $n\to \infty$.

Remarks:

• We can think of an ODE $\frac{dx}{dt}=x$ as a difference equation $x_{n+1}=x_n+x_n\delta t$ in the limit of $\delta t\to 0$.
• For ODEs, it is typically easier to compute using a continuous picture.

Continuum Limit of Difference Equations

Consider the difference equation $x_n\in \mathbb{R}$ with

$$x_{n+1}=x_n+\delta tu,x_0=0,n=0,1,2,\dots$$

does this have a continuum limit? In particular, is there an $x(t)$ such that

$$x_n\to x(t)\text{ as }n\to \infty ,\delta t\to 0\text{ with }n\cdot \delta t=t?$$

Check $x_1=\delta tu,\dots ,x_n=n\delta tu=ut$. So we approach $ut$. Thus, the continuum limit is

$$x(t)=ut,\frac{dx}{dt}=u$$

Remark: Suppose we took

$$x_{n+1}=x_n+\delta t^{\alpha }u$$

then

$$x_n=n\delta t^{\alpha }u=t\cdot \delta t^{\alpha -1}u$$

so that

$$x_n\to \{\begin{array}{ll}0& \alpha >1\\ \infty & \alpha <1\end{array}$$

as $\delta t\to 0,n\to \infty ,n\delta t=t$. This implies there is no continuum limit unless $\alpha =1$.

We essentially are reducing the discrete step-wise system to an analytical model. It is a formal operation to recover a differential equation from a difference equation.

Intuitively, this is subdividing some interval of time $t$ into $n$ parts of size $\delta$, such that $n(\delta t)=t$. Non-existence of an $\alpha$ means that further subdivision diverges to infinity or collapses identically to $0$. In particular, we would be unable to map a discrete model to a continuous differential framework.